The following is my code.
First come a special case where n is 1.
If n is not 1, more work should be done.

First, the nails are sorted based on their x-coordination.
The leftmost nail is accessed, and compared to the second leftmost nail.
With some boundary case, all the nails can be sorted along the clockwise or counter-clockwise direction.

Finally, I measure the distances between adjacent nails with pythagorean theorm.
Together with 2*pi*r, which accounts for the round edges, should be the answer.
What exactly does the code fail Test #5?


import math

def nail(n, r):
    return 2 * math.pi * r

def pyth(a, b):
    return math.sqrt(a**2 + b**2)

def perimeter(coord):
    ans = 0.0
    for i in range(len(coord)):
        ans += pyth((coord[i][0] - coord[i-1][0]), (coord[i][1] - coord[i-1][1]))
    return ans

if __name__ == "__main__":
    n, r = raw_input().split()
    n = int(n)
    r = float(r)
    coord = []
    for each in range(n):
        i, j = raw_input().split()
        coord += [(float(i), float(j))]

    if n == 1:
        print "%.2f" % (2 * math.pi * r)

    else:
        coord.sort()
        directed_coord = []
        directed_coord += [coord.pop(0)]
        if coord[0][1] > directed_coord[0][1]:
            clockwise = True
            directed_coord += [coord.pop(0)]
        else:
            clockwise = False
            directed_coord += [coord.pop(0)]

        if clockwise == True:
            for i in coord[:]:
                if i[1] > directed_coord[-1][1]:
                    coord.remove(i)
                    directed_coord += [i]
            for i in coord[:]:
                if i[1] >= coord[-1][1]:
                    coord.remove(i)
                    directed_coord += [i]
            coord.reverse()
            directed_coord += coord
        else:
            for i in coord[:]:
                if i[1] < directed_coord[-1][1]:
                    coord.remove(i)
                    directed_coord += [i]
            for i in coord[:]:
                if i[1] <= coord[-1][1]:
                    coord.remove(i)
                    directed_coord += [i]
            coord.reverse()
            directed_coord += coord

        ans = nail(n, r) + perimeter(directed_coord)
        print "%.2f" % ans